A model-constraining objective functional for geophysical joint inversion based on logistic function types

We propose a model-constraining objective function to be used in geophysical joint inversions, referred to briefly as semblance. The semblance formulation is based on the idea of logistic functions, where a binary dependent variable adds non-zero or zero accumulation terms for the, respectively, passing or failing of preset anomaly thresholds with respect to a known background model. Similar to other model-constraining functionals, such as structural constraints imposed by a cross-gradient function, the semblance involves a model pair (m₁,m₂) for which structural similarity is calculated, here by means of logistic functions. The latter return semblance values in the range [0,1] for minimal and maximal semblance, respectively. Maximal semblance occurs when both m₁ is within [a₁,b₁] and m₂ is within [a₂,b₂], where [a,b] are predefined anomaly thresholds.

The figure demonstrates the concept by means of an inversion without any data input and only the semblance constraint active. The reference (a) is the model to invert for, which would be the anomalous property distribution m₁ in a joint inversion for an unknown m₂, where the degree of anomaly is measured with respect to the background (b). The image (c) is the reconstruction of the anomaly using the semblance criteria [a₂,b₂]=[0,1300] (in m/s), and a homogeneous starting model (v=2000 m/s). Multiple logistic function terms can be combined in this approach. For example, time-lapse data inversions can employ an additional logistic function designed to rate relative property changes with respect to the baseline model.